Pascal like Triangle, Sierpinski like
Gaskets and Fibonacci like Sequences.
- New mathematical theorems discovered by high school students.-
Tomo Hamada
Tomohide Hashiba
Hiroshi Matsui
Naoki Saida
Toshiyuki Yamauchi
          Proofs part 1
          Proofs part 2
          Proofs part 3
Here we are going to present some new mathematical theorems that we discovered using computers. A part of this discovery has already been published in mathematics magazines, and new theorems will be introduced here.
We made Pascal-like triangles that are generalizations of the well known Pascal's triangle.
Let's introduce the game we used to get the triangles.
Let p, n, m be fixed natural number such that m≤n. We have players θ₁,θ₂,θ₃… , θp who are seated around a circle. The game begins with player θ. Proceeding in order, a box is passed from hand to hand. The box contains n numbers. The numbers in the box are assigned in numerical order, from 1 to n. The numbers 1, 2, ...,m are printed in red.
When a player receives the box, he draws a number from it. The players cannot see the numbers when they draw them, and hence they draw at random. Once a number is drawn, that number will not be returned to the box. If any player gets a number printed in red, he is the loser and the game ends.
Let F(p,n,m,v) be the vth player's losing in the game. Then for fixed number p and v with v≤p the list {F(p,n,m,v), n=1,2,... and m ≤n} forms a Pascal-like triangle.
With these triangles we discovered Fibonacci like sequences, Sierpinski like gaskets and interesting relations between these sequences and the Fibonacci sequences.
These new facts are the creation of a combination of the power of computers and the creativity of high school students!